Abstract: A d-contraction is a d-tuple (T1, . . . , Td) of mutually commuting operators acting on a common Hilbert space H such that ‖T1ξ1 + T2ξ2 + · · · + Tdξd‖ ≤ ‖ξ1‖ + ‖ξ2‖ + · · · + ‖ξd‖ for all ξ1, ξ2, . . . , ξd ∈ H. These are the higher dimensional counterparts of contractions. We show that many of the operator-theoretic aspects of function theory in the unit disk generalize to the unit ball Bd in complex d-space, including von Neumann’s inequality and the model theory of contractions. These results depend on properties of the d-shift, a distinguished d-contraction which acts on a new H2 space associated with Bd, and which is the higher dimensional counterpart of the unilateral shift. H 2 and the d-shift are highly unique. Indeed, by exploiting the noncommutative Choquet boundary of the d-shift relative to its generated C∗-algebra we find that there is more uniqueness in dimension d ≥ 2 than there is in dimension one.

Abstract: Pseudo-Differential Operators. Mellin Pseudo-Differential Operators on Manifolds with Conical Singularities. Pseudo-Differential Calculus on Manifolds with Edges. Boundary Value Problems. Bibliography. Index.

Abstract: Introduction Basic Notions Generalized Solutions to Linear Partial Differential Equations Generalized Pseudodifferential Operators Bibliography

Abstract: Introduction: Euclidean linear spaces Orthogonal and unitary linear transformations Orthogonal and unitary transformations. Singular values Matrices of operators in the Euclidean space: Unitary similar transformations. The Schur theorem Alternation theorems The Weyl inequalities Variational principles Resolvent and dichotomy of spectrum Quadratic forms in the spectrum dichotomy problem Matrix equations and projections The Hausdorff set of a matrix Application of spectral analysis. The most important algorithms: Matrix operators as models of differential operators Application of the theory of functions of complex variable Computational algorithms of spectral analysis Bibliography Index.

Abstract: We prove that the spectrum of certain non-self-adjoint Schrodinger operators is unstable in the semi-classical limit. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum.

Abstract: In this paper we give a further development of the results of the paper [1] and apply it to convolution operators in the spaces Lp(x) . We consider the question of extendability of the Young theorem: well known for constant p and q, to the case when they may be variable. We also potential type operators with the Kernel . In section 1 we develop some estimates for Lp(x) -norms of power functions of distance truncated to exterior of a ball of radius . Section 2 deals with convolution operators in the spaces Lp(x) and Section 3 is devoted to potential type operators.

Abstract: This paper is the continuation of [13]. Here we apply the Fourier Integral Operators (FIOs) calculus developed therein to the solution of Cauchy problems for a class of hyperbolic linear systems and operators. We require, in particular, that the eigenvalues of the principal part of the symbol of the operator matrix (or the roots of the characteristic equation associated to the operator of order ν) satisfy suitable separation conditions at the infinity. *** DIRECT SUPPORT *** A00BP044 00004

Abstract: Background. The Main Tools. The Irreducibility and Strong Irreducibility of Operators. The Strongly Irreducible Operator in Some Classes of Operators. The Spectral Pictures of Strongly Irreducible Operators and the Approximate Jordan Decomposition Theorem. Compact Perturbations of Strongly Irreducible Operators. Strongly Irreducible Operators in Nest Algebras. Some Other Results about Strongly Irreducible Operators. References.

Abstract: One of the great challenges of the spectral theory of singular integral operators is a theory unifying the three "forces" which determine the local spectra: the oscillation of the Carleson curve, the oscillation of the Muckenhoupt weight, and the oscillation of the coecients. In this paper we demonstrate how by employing the method of limit operators one can describe the spectra in case all data of the operator (the curve, the weight, and the coecients) are slowly oscillating.

Abstract: The authors prove that bilinear operators given by finite sums of products of Calderon-Zygmund operators on Rn are bounded from HK11 q1 × HK22 q2 into HK α,p q if and only if they have vanishing moments up to a certain order dictated by the target space. HereHK q are homogeneous Herztype Hardy spaces with 1/p = 1/p1 + 1/p2, 0 < pi ≤ ∞, 1/q = 1/q1 + 1/q2, 1 < q1, q2 < ∞, 1 ≤ q < ∞, α = α1 + α2 and −n/qi < αi < ∞. As an application they obtain that the commutator of a Calderon-Zygmund operator with a BMO function maps a Herz space into itself.

Abstract: This paper deals with periodic pseudodifferential operators on the unit circle T' = R'/Z 1 . The main results are asymptotic expansions for adjoints and products of periodic pseudodifferential operators.

Abstract: Wave Scattering in 1-D Nonconservative Media T. Aktosun, et al. Resolvent Estimates for Schroedinger-type and Maxwell Equations with Applications M. Ben-Artzi, J. Nemirovsky. Symmetric Solutions of Ginzburg-Landau Equations S. Gustafson. Quantum Mechanics and Relativity: Their Unification by Local Time H. Kitada. On Embedded Eigenvalues of Perturbed Periodic Schroedinger Operators P. Kuchment, B. Vainberg. On Principal Eigenvalues for Indefinite-Weight Elliptic Problems Y. Pinchover. Scattering by Obstacles in Acoustic Waveguides A.G. Ramm, G.N. Makrakis. Recovery of Compactly Supported Spherically Symmetric Potentials from the Phase Shift of the s-Wave A.G. Ramm. A Turning Point Problem Arising in Connection with a Limiting Absorption Principle for Schroedinger Operators with Generalized Von Neumann-Wigner Potentials P. Rejto, M. Taboada. Eigenvalue Problems for Semilinear Equations M. Schechter. Spectral Operators Generated by 3-Dimensional Damped Wave Equation and Applications to Control Theory M.A. Shubov. Invertibility of Nonlinear Operators and Parameter Continuation Method V.A. Trenogin. Sturm-Liouville Differential Operators with Singularities V.Yurko. Index.

Abstract: Spaces and Operators. Linear Semigroups. Semilinear Problems. Dissipative Operators. Nonlinear Semigroups. Appendices. References. Indexes.

Abstract: Preface CHAPTER 1: REGULAR EXTENSIONS Linear Operators Spectrum of a Linear Operator Hermitian Operators Symmetric Operators Regular Extensions of Hermitian Operators Dissipative Extensions of Hermitian Operators Accretive Operators CHAPTER 2: REGULAR EXTENSIONS WITH RESTRICTIONS Self-Adjoint Bound-Preserving Extensions of Semibounded Symmetric Operators. Theorem and Hypothesis of Von Neumann Proof of the Von Neumann Hypothesis (Friedrichs Method) Self-Adjoint Norm-Preserving Extensions of Hermitian Contractions Normal Norm-Preserving Extensions of Hermitian Contractions Krein's Proof of the Von Neumann Hypothesis Squared Symmetric Operators Self-Adjoint Bound-Preserving Extensions of Semibounded Hermitian Operators Regular U-Invariant Extensions of Hermitian Operators Canonical Dissipative Extensions of Hermitian Operators CHAPTER 3: BOUNDARY-VALUE SPACES OF HERMITIAN OPERATORS Definition and General Properties of Boundary-Value Spaces Description of Regular-Extensions of Hermitian Operators in Terms of Boundary-Value Spaces Characteristic Functions of Hermitian Operators Spectral Properties of Regular Extensions CHAPTER 4: EXAMPLES AND APPLICATIONS Quasidifferential Operators Boundary-Value Spaces for Model of Zero-Range Potentials with Internal Structure Some Properties of Nonperturbed Operators Abstract Wave Equation Elements of the Lax-Phillips Scattering Theory for -Perturbed Abstract Wave Equation References Subject Index Notation

Abstract: Some principles of the operator theory in a linear space with two norms are established in this paper. The well-known Hilbert-Schmidt theorem on the eigenfunction expansion of sourcewise represented functions, Mercer's theorem and other results can be consider as special cases of the statements presented. The general approach proposed is used to construct the theory of symmetrizable operators and to investigate the asymptotic behaviour of eigenvalues of compact operators.

Abstract: We extend previous work on conformally covariant differential operators to consider the case of second-order operators acting on symmetric traceless tensor fields. The corresponding flat space Green function is explicitly constructed and shown to be in accord with the requirements of conformal invariance.

Abstract: Beilinson Completion Algebras (BCAs) are generalizations of complete local rings, and have a rich algebraic-analytic structure. These algebras were introduced in my paper "Traces and Differential Operators over Beilinson Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are used to give an explicit construction of the Grothendieck residue complex on an algebraic scheme. This construction reveals new properties of the residue complex, and in particular its interaction with differential operators. Applications include: (i) results on the algebraic structure of rings of differential operators; (ii) an analysis of the niveau spectral sequence of De Rham homology; (iii) a proof of the contravariance of De Rham homology w.r.t. etale morphisms; (iv) an algebraic description of the intersection cohomology D-module of a curve.

Abstract: Using positivity, abstract extrapolation spaces, and a generalization of the Jorgens–Vidav-Voigt perturbation theorem for the essential spectral radius of strongly continuous semigroups given in [19], we study the asymptotic behavior of a linear age-dependent population problem with spatial diffusion where the birth and the death rates depend on the spatial variable.

Abstract: As a central topic certain relations between operator matrices are investigated which are called delta relations. The main aim of these relations is to reduce questions about classes of operators without invertibility symbol to those which admit an invertibility symbol. Particular attention is devoted to the generalized inversion of such operators. Different kinds of relations are introduced in order to analyze the \"information\" contained in the symbols of the related operators. Several examples are considered and the theory is also applied to singular integral operators with Carleman shift. Asymptotic solutions of equations characterized by those operators are presented. The approach simplifies several known results, makes the theory more rigorous from the operator theoretic point of view, and allows further conclusions in a very compact form.

Abstract: Part I: Generalized Functions and Operator Theory An Introduction to Hyperfunctions and ?-expansions, G Lumer Partial Inner Product Spaces of Analytic Functions, J-P Antoine Rigged Spectral States: A Proclivity for Eigenvalues, K Gustafson Densities of Singular Measures and Generalized Spectral Decompositions, I Antoniou and Z Suchanecki Convolution Kernels and Generalized Functions, B Baumer, G Lumer, and F Neubrander Spectral Theory of Closed Linear Operators on Banach Spaces from a Locally Convex Point of View, V Wrobel Ultradistributions and the Levinson Condition, I Cioranescu and L Zsido Representation of the Derivatives and Products of the Delta Function in Hilbert Space, Yu Melnikov Series Representations of the Complex Delta Function, I Antoniou, Z Suchanecki, and S Tasaki Antieigenvalues: An Extended Spectral Theory, K Gustafson Part II: Operator Theory and Dynamical Systems Laws of Nature, Probability, and Time Symmetry Breaking, I Prigogine and T Petrosky Extended Spectral Decompositions of Evolution Operators, I Antoniou and S Shkarin Some Little Things About Rigged Hilbert Spaces and Quantum Mechanics and All That, A Bohm, M Gadella, and S Wickramasekara Axiomatics of Thermodynamics and Quantum Chaos, V Maslov Stochastic Evolution on Product Manifolds S Albeverio, A Daletskii, and Yu Kondratiev Interaction Problems with Distributions and Hyperfunctions Data, G Lumer Absolute Continuity of Convolutions of Singular Measures and New Branches of Spectrum of Liouvillians and Few-Body Hamiltonians, L Bos and B Pavlov On Scattering Theories Involving Moving Boundaries, G F Roach The Eigenvalue Problem for Networks of Beams, B Dekoninck and S Nicaise Generalized Perturbations and Operator Relations, P Kurasov and B Pavlov On Spectral Analysis of a Class of Integral-Difference Collision Operators, Yu Melnikiv Dynamical Aspects of Processes with Long-Range Memory, M Courbage

Abstract: Suppose that K is a closed, total cone in a real Banach space X, that A :X!X is a bounded linear operator which maps K into itself, and that A« denotes the Banach space adjoint of A. Assume that r, the spectral radius of A, is positive, and that there exist x ! 1 0 and m& 1 with Am(x ! ) ̄ rmx ! (or, more generally, that there exist x ! a (®K ) and m& 1 with Am(x ! )& rmx ! ). If, in addition, A satisfies some hypotheses of a type used in mean ergodic theorems, it is proved that there exist u `K®20 ́ and θ `K «®20 ́ with A(u) ̄ ru, A«(θ) ̄ rθ and θ(u)\" 0. The support boundary of K is used to discuss the algebraic simplicity of the eigenvalue r. The relation of the support boundary to H. Schaefer’s ideas of quasi-interior elements of K and irreducible operators A is treated, and it is noted that, if dim(X )\" 1, then there exists an x `K®20 ́ which is not a quasi-interior point. The motivation for the results is recent work of Toland, who considered the case in which X is a Hilbert space and A is self-adjoint ; the theorems in the paper generalize several of Toland’s propositions.